What is the fractional part function of $e^x$?

Question

Given a real positive number $x\in\mathbb{R^+}$.

What is the function of the fractional part of $e^x$?

Answer 1

As Johannes Kloos suggests, you can use the Taylor expansion. Here's the straightforward application; there may be better ones.

Let s = 0 and t = 1.
Start a loop with n = 1, incrementing by 1 each time.
  Let t = t * x / n.
  If t is sufficiently small, exit the loop.
  Let s = s + (t - floor(t)).
  If s > 1, let s = s - 1.
Return t.

Essentially all the rounding error comes from the t - floor(t). Some numerical analysis should give the maximum size of t and hence the number of significant digits in t - floor(t); this, in turn, gives an idea of what "sufficiently small" means.

Answer 2

The Fractional part of $e^x$ is given by $\text{frac}(e^x)=e^x-\lfloor e^x\rfloor.$

Answer 3

Suppose $x$ is between $100$ and $101$. To compute even one significant figure for the fractional part of $e^x$, you will need to know $x$ accurate to about $44$ significant figures...